GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 4.2s

Alternatives: 16

Speedup: 0.9×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t\_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (let* ((t_0 (- (* alpha alpha) 1.0)))
  (/
   t_0
   (*
    (* PI (log (* alpha alpha)))
    (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((t_0 * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	return Float32(t_0 / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(t_0 * cosTheta) * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) - single(1.0);
	tmp = t_0 / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t\_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (let* ((t_0 (- (* alpha alpha) 1.0)))
  (/
   t_0
   (*
    (* PI (log (* alpha alpha)))
    (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((t_0 * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	return Float32(t_0 / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(t_0 * cosTheta) * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) - single(1.0);
	tmp = t_0 / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Alternative 1: 98.7% accurate, 0.8× speedup

Math

\[\frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right)}\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 (- (* alpha alpha) 1.0)
 (log
  (pow
   (* alpha alpha)
   (* (fma (* cosTheta (fma alpha alpha -1.0)) cosTheta 1.0) PI)))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / logf(powf((alpha * alpha), (fmaf((cosTheta * fmaf(alpha, alpha, -1.0f)), cosTheta, 1.0f) * ((float) M_PI))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / log((Float32(alpha * alpha) ^ Float32(fma(Float32(cosTheta * fma(alpha, alpha, Float32(-1.0))), cosTheta, Float32(1.0)) * Float32(pi)))))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   5. lift-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right) \cdot \color{blue}{\log \left(\alpha \cdot \alpha\right)}} \]

   6. log-pow-rev N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right)}\right)}} \]

   7. lower-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right)}\right)}} \]

   8. lower-pow.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\alpha \cdot \alpha\right)}^{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right)}\right)}} \]

   9. lower-*.f32 98.7%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\color{blue}{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right)}}\right)} \]

3. Applied rewrites 98.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\left(\alpha \cdot \alpha\right)}^{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right)}\right)}} \]
4. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\left(1 - \alpha \cdot \alpha\right) \cdot \frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\mathsf{fma}\left(\pi \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right), cosTheta, \pi\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (*
 (- 1.0 (* alpha alpha))
 (/
  (/ -1.0 (log (* alpha alpha)))
  (fma (* PI (* cosTheta (fma alpha alpha -1.0))) cosTheta PI))))

C

float code(float cosTheta, float alpha) {
	return (1.0f - (alpha * alpha)) * ((-1.0f / logf((alpha * alpha))) / fmaf((((float) M_PI) * (cosTheta * fmaf(alpha, alpha, -1.0f))), cosTheta, ((float) M_PI)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(1.0) - Float32(alpha * alpha)) * Float32(Float32(Float32(-1.0) / log(Float32(alpha * alpha))) / fma(Float32(Float32(pi) * Float32(cosTheta * fma(alpha, alpha, Float32(-1.0)))), cosTheta, Float32(pi))))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   3. mult-flip N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   5. lift--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha - 1\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)} \]

   6. sub-negate-rev N/A

      \[\leadsto \color{blue}{\left(1 - \alpha \cdot \alpha\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)} \]

   7. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(1 - \alpha \cdot \alpha\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)\right)} \]

   9. distribute-neg-frac N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   10. lower-/.f32 N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   11. metadata-eval 98.4%

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   12. lift-*.f32 N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   13. *-commutative N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

3. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{-1}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   3. *-commutative N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi}} \]

   5. lower-/.f32 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi}} \]

   6. lower-/.f32 98.3%

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\color{blue}{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi} \]

   7. lift-*.f32 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\color{blue}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi}} \]

   8. *-commutative N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\color{blue}{\pi \cdot \mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}} \]

   9. lift-fma.f32 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{\left(\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right) \cdot cosTheta + 1\right)}} \]

   10. distribute-lft-in N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\color{blue}{\pi \cdot \left(\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right) \cdot cosTheta\right) + \pi \cdot 1}} \]

5. Applied rewrites 98.3%

   \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\mathsf{fma}\left(\pi \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right), cosTheta, \pi\right)}} \]
6. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \pi} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 (- (* alpha alpha) 1.0)
 (*
  (*
   (fma (* cosTheta (fma alpha alpha -1.0)) cosTheta 1.0)
   (log (* alpha alpha)))
  PI)))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((fmaf((cosTheta * fmaf(alpha, alpha, -1.0f)), cosTheta, 1.0f) * logf((alpha * alpha))) * ((float) M_PI));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(fma(Float32(cosTheta * fma(alpha, alpha, Float32(-1.0))), cosTheta, Float32(1.0)) * log(Float32(alpha * alpha))) * Float32(pi)))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right) \cdot \pi}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right) \cdot \pi}} \]

3. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \pi}} \]
4. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 (fma alpha alpha -1.0)
 (*
  (* (fma (* cosTheta (fma alpha alpha -1.0)) cosTheta 1.0) PI)
  (log (* alpha alpha)))))

C

float code(float cosTheta, float alpha) {
	return fmaf(alpha, alpha, -1.0f) / ((fmaf((cosTheta * fmaf(alpha, alpha, -1.0f)), cosTheta, 1.0f) * ((float) M_PI)) * logf((alpha * alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(fma(Float32(cosTheta * fma(alpha, alpha, Float32(-1.0))), cosTheta, Float32(1.0)) * Float32(pi)) * log(Float32(alpha * alpha))))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)\right)\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha - 1\right)}\right)\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. sub-flip N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. distribute-neg-in N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)}\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right) + \color{blue}{1}\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. distribute-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\alpha \cdot \alpha\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. sqr-abs-rev N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left|\alpha\right| \cdot \left|\alpha\right|}\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left|\alpha\right| \cdot \left(\mathsf{neg}\left(\left|\alpha\right|\right)\right)}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left|\alpha\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|\alpha\right|\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   11. sqr-neg-rev N/A

       \[\leadsto \frac{\color{blue}{\left|\alpha\right| \cdot \left|\alpha\right|} + \left(\mathsf{neg}\left(1\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   12. sqr-abs-rev N/A

       \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha} + \left(\mathsf{neg}\left(1\right)\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   13. lower-fma.f32 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, \mathsf{neg}\left(1\right)\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   14. metadata-eval 98.4%

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   16. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   17. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   18. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

3. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]
4. Add Preprocessing

Alternative 5: 97.5% accurate, 1.2× speedup

Math

\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 (- (* alpha alpha) 1.0)
 (*
  (* PI (log (* alpha alpha)))
  (+ 1.0 (* (* -1.0 cosTheta) cosTheta)))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((-1.0f * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(Float32(-1.0) * cosTheta) * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((single(-1.0) * cosTheta) * cosTheta)));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 97.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot \color{blue}{cosTheta}\right) \cdot cosTheta\right)} \]

4. Applied rewrites 97.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 1.3× speedup

Math

\[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \pi} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 (- (* alpha alpha) 1.0)
 (* (* (fma (- cosTheta) cosTheta 1.0) (log (* alpha alpha))) PI)))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((fmaf(-cosTheta, cosTheta, 1.0f) * logf((alpha * alpha))) * ((float) M_PI));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(fma(Float32(-cosTheta), cosTheta, Float32(1.0)) * log(Float32(alpha * alpha))) * Float32(pi)))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 97.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot \color{blue}{cosTheta}\right) \cdot cosTheta\right)} \]

4. Applied rewrites 97.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)\right) \cdot \pi}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)\right) \cdot \pi}} \]

6. Applied rewrites 97.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \pi}} \]
7. Add Preprocessing

Alternative 7: 97.5% accurate, 1.3× speedup

Math

\[\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(\left(\pi \cdot \log \alpha\right) \cdot 2\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 (fma alpha alpha -1.0)
 (* (fma (- cosTheta) cosTheta 1.0) (* (* PI (log alpha)) 2.0))))

C

float code(float cosTheta, float alpha) {
	return fmaf(alpha, alpha, -1.0f) / (fmaf(-cosTheta, cosTheta, 1.0f) * ((((float) M_PI) * logf(alpha)) * 2.0f));
}

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(fma(Float32(-cosTheta), cosTheta, Float32(1.0)) * Float32(Float32(Float32(pi) * log(alpha)) * Float32(2.0))))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 97.5%

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot \color{blue}{cosTheta}\right) \cdot cosTheta\right)} \]

4. Applied rewrites 97.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
5. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha - 1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha} - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. difference-of-sqr-1 N/A

      \[\leadsto \frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. difference-of-sqr--1 N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha + -1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. lift-fma.f32 97.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   8. lower-*.f32 97.4%

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   9. lift-+.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(1 + \left(-1 \cdot cosTheta\right) \cdot cosTheta\right)} \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\left(-1 \cdot cosTheta\right) \cdot cosTheta + 1\right)} \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\color{blue}{\left(-1 \cdot cosTheta\right) \cdot cosTheta} + 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   12. lower-fma.f32 97.4%

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\mathsf{fma}\left(-1 \cdot cosTheta, cosTheta, 1\right)} \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-1 \cdot \color{blue}{cosTheta}, cosTheta, 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(\mathsf{neg}\left(cosTheta\right), cosTheta, 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   15. lower-neg.f32 97.4%

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   16. lower-neg.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f32, \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)\right)} \]

   17. lower-neg.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(\pi \cdot \mathsf{Rewrite=>}\left(lift-log.f32, \log \left(\alpha \cdot \alpha\right)\right)\right)} \]

   18. lower-neg.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(\pi \cdot \log \mathsf{Rewrite=>}\left(lift-*.f32, \left(\alpha \cdot \alpha\right)\right)\right)} \]

   19. lower-neg.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(\pi \cdot \log \mathsf{Rewrite=>}\left(pow2, \left({\alpha}^{2}\right)\right)\right)} \]

6. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(-cosTheta, cosTheta, 1\right) \cdot \left(\left(\pi \cdot \log \alpha\right) \cdot 2\right)}} \]
7. Add Preprocessing

Alternative 8: 95.2% accurate, 1.3× speedup

Math

\[\frac{1}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot 1}{1 - \alpha \cdot \alpha} \cdot \left(-\pi\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/
 1.0
 (* (/ (* (log (* alpha alpha)) 1.0) (- 1.0 (* alpha alpha))) (- PI))))

C

float code(float cosTheta, float alpha) {
	return 1.0f / (((logf((alpha * alpha)) * 1.0f) / (1.0f - (alpha * alpha))) * -((float) M_PI));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(1.0) / Float32(Float32(Float32(log(Float32(alpha * alpha)) * Float32(1.0)) / Float32(Float32(1.0) - Float32(alpha * alpha))) * Float32(-Float32(pi))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(1.0) / (((log((alpha * alpha)) * single(1.0)) / (single(1.0) - (alpha * alpha))) * -single(pi));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi}}{\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   6. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}{\frac{\alpha \cdot \alpha - 1}{\pi}}}} \]

   7. lower-unsound-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}{\frac{\alpha \cdot \alpha - 1}{\pi}}}} \]

   8. lower-unsound-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}{\frac{\alpha \cdot \alpha - 1}{\pi}}}} \]

3. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}}} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\color{blue}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}}} \]

   3. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\color{blue}{\alpha \cdot \alpha + -1}}{\pi}}} \]

   4. difference-of-sqr--1 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}}{\pi}}} \]

   5. difference-of-sqr-1 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\color{blue}{\alpha \cdot \alpha - 1}}{\pi}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\color{blue}{\alpha \cdot \alpha} - 1}{\pi}}} \]

   7. sub-negate-rev N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\color{blue}{\mathsf{neg}\left(\left(1 - \alpha \cdot \alpha\right)\right)}}{\pi}}} \]

   8. lift--.f32 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - \alpha \cdot \alpha\right)}\right)}{\pi}}} \]

   9. distribute-frac-neg N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\color{blue}{\mathsf{neg}\left(\frac{1 - \alpha \cdot \alpha}{\pi}\right)}}} \]

   10. distribute-frac-neg2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{\color{blue}{\frac{1 - \alpha \cdot \alpha}{\mathsf{neg}\left(\pi\right)}}}} \]

   11. associate-/r/ N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{1 - \alpha \cdot \alpha} \cdot \left(\mathsf{neg}\left(\pi\right)\right)}} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \log \left(\alpha \cdot \alpha\right)}{1 - \alpha \cdot \alpha} \cdot \left(\mathsf{neg}\left(\pi\right)\right)}} \]

5. Applied rewrites 98.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}{1 - \alpha \cdot \alpha} \cdot \left(-\pi\right)}} \]

6. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot \color{blue}{1}}{1 - \alpha \cdot \alpha} \cdot \left(-\pi\right)} \]
7. Step-by-step derivation

8. Applied rewrites 95.0%

   \[\leadsto \frac{1}{\frac{\log \left(\alpha \cdot \alpha\right) \cdot \color{blue}{1}}{1 - \alpha \cdot \alpha} \cdot \left(-\pi\right)} \]
9. Add Preprocessing

Alternative 9: 95.1% accurate, 1.6× speedup

Math

\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) 1.0)))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * 1.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(1.0)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) * log((alpha * alpha))) * single(1.0));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.2%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
5. Add Preprocessing

Alternative 10: 95.1% accurate, 1.6× speedup

Math

\[\left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (* (- 1.0 (* alpha alpha)) (/ -1.0 (* PI (log (* alpha alpha))))))

C

float code(float cosTheta, float alpha) {
	return (1.0f - (alpha * alpha)) * (-1.0f / (((float) M_PI) * logf((alpha * alpha))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(1.0) - Float32(alpha * alpha)) * Float32(Float32(-1.0) / Float32(Float32(pi) * log(Float32(alpha * alpha)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(1.0) - (alpha * alpha)) * (single(-1.0) / (single(pi) * log((alpha * alpha))));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   3. mult-flip N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\alpha \cdot \alpha - 1\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   5. lift--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha - 1\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)} \]

   6. sub-negate-rev N/A

      \[\leadsto \color{blue}{\left(1 - \alpha \cdot \alpha\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)} \]

   7. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(1 - \alpha \cdot \alpha\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)\right)} \]

   9. distribute-neg-frac N/A

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   10. lower-/.f32 N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   11. metadata-eval 98.4%

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   12. lift-*.f32 N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   13. *-commutative N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

3. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\left(\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right) \cdot \pi\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

4. Taylor expanded in cosTheta around 0

   \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\pi} \cdot \log \left(\alpha \cdot \alpha\right)} \]
5. Step-by-step derivation

   1. lower-PI.f32 95.1%

      \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} \]

6. Applied rewrites 95.1%

   \[\leadsto \left(1 - \alpha \cdot \alpha\right) \cdot \frac{-1}{\color{blue}{\pi} \cdot \log \left(\alpha \cdot \alpha\right)} \]
7. Add Preprocessing

Alternative 11: 95.0% accurate, 1.8× speedup

Math

\[\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ (/ (fma alpha alpha -1.0) (log (* alpha alpha))) PI))

C

float code(float cosTheta, float alpha) {
	return (fmaf(alpha, alpha, -1.0f) / logf((alpha * alpha))) / ((float) M_PI);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / log(Float32(alpha * alpha))) / Float32(pi))
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \pi\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \left(\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

3. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}} \]

4. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{1}} \]
5. Step-by-step derivation

6. Applied rewrites 95.1%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{1}} \]

7. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\color{blue}{\pi}} \]
8. Step-by-step derivation

   1. lower-PI.f32 95.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi} \]

9. Applied rewrites 95.1%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\color{blue}{\pi}} \]
10. Add Preprocessing

Alternative 12: 65.7% accurate, 1.9× speedup

Math

\[\frac{\frac{-1}{\pi}}{1 \cdot \log \left(\alpha \cdot \alpha\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ (/ -1.0 PI) (* 1.0 (log (* alpha alpha)))))

C

float code(float cosTheta, float alpha) {
	return (-1.0f / ((float) M_PI)) / (1.0f * logf((alpha * alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-1.0) / Float32(pi)) / Float32(Float32(1.0) * log(Float32(alpha * alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) / single(pi)) / (single(1.0) * log((alpha * alpha)));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.2%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
6. Step-by-step derivation

7. Applied rewrites 65.6%

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{-1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{-1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{-1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot 1} \]

   4. associate-*l* N/A

      \[\leadsto \frac{-1}{\color{blue}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot 1\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{\pi}}{\log \left(\alpha \cdot \alpha\right) \cdot 1}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{\pi}}{\log \left(\alpha \cdot \alpha\right) \cdot 1}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{\pi}}}{\log \left(\alpha \cdot \alpha\right) \cdot 1} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{\pi}}{\color{blue}{1 \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   9. lower-*.f32 65.7%

      \[\leadsto \frac{\frac{-1}{\pi}}{\color{blue}{1 \cdot \log \left(\alpha \cdot \alpha\right)}} \]

9. Applied rewrites 65.7%

   \[\leadsto \color{blue}{\frac{\frac{-1}{\pi}}{1 \cdot \log \left(\alpha \cdot \alpha\right)}} \]
10. Add Preprocessing

Alternative 13: 65.6% accurate, 1.9× speedup

Math

\[\frac{\frac{-1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot 1} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ (/ -1.0 (log (* alpha alpha))) (* PI 1.0)))

C

float code(float cosTheta, float alpha) {
	return (-1.0f / logf((alpha * alpha))) / (((float) M_PI) * 1.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-1.0) / log(Float32(alpha * alpha))) / Float32(Float32(pi) * Float32(1.0)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) / log((alpha * alpha))) / (single(pi) * single(1.0));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \pi\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \left(\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

3. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}} \]

4. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{1}} \]
5. Step-by-step derivation

6. Applied rewrites 95.1%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{1}} \]

7. Taylor expanded in alpha around 0

   \[\leadsto \frac{\frac{\color{blue}{-1}}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot 1} \]
8. Step-by-step derivation

9. Applied rewrites 65.6%

   \[\leadsto \frac{\frac{\color{blue}{-1}}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot 1} \]
10. Add Preprocessing

Alternative 14: 65.6% accurate, 2.0× speedup

Math

\[\frac{-1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ -1.0 (* (* PI (log (* alpha alpha))) 1.0)))

C

float code(float cosTheta, float alpha) {
	return -1.0f / ((((float) M_PI) * logf((alpha * alpha))) * 1.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(-1.0) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(1.0)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-1.0) / ((single(pi) * log((alpha * alpha))) * single(1.0));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.2%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
6. Step-by-step derivation

7. Applied rewrites 65.6%

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
8. Add Preprocessing

Alternative 15: 65.6% accurate, 2.2× speedup

Math

\[\frac{\frac{-0.5}{\log \alpha}}{\pi \cdot 1} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ (/ -0.5 (log alpha)) (* PI 1.0)))

C

float code(float cosTheta, float alpha) {
	return (-0.5f / logf(alpha)) / (((float) M_PI) * 1.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-0.5) / log(alpha)) / Float32(Float32(pi) * Float32(1.0)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-0.5) / log(alpha)) / (single(pi) * single(1.0));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \left(\alpha \cdot \alpha\right) \cdot \pi\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \left(\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

3. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}} \]

4. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{1}} \]
5. Step-by-step derivation

6. Applied rewrites 95.1%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi \cdot \color{blue}{1}} \]

7. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2}}{\log \alpha}}}{\pi \cdot 1} \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{-1}{2}}{\color{blue}{\log \alpha}}}{\pi \cdot 1} \]

   2. lower-log.f32 65.6%

      \[\leadsto \frac{\frac{-0.5}{\log \alpha}}{\pi \cdot 1} \]

9. Applied rewrites 65.6%

   \[\leadsto \frac{\color{blue}{\frac{-0.5}{\log \alpha}}}{\pi \cdot 1} \]
10. Add Preprocessing

Alternative 16: 65.6% accurate, 2.4× speedup

Math

\[\frac{-1}{2 \cdot \left(\pi \cdot \log \alpha\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  (/ -1.0 (* 2.0 (* PI (log alpha)))))

C

float code(float cosTheta, float alpha) {
	return -1.0f / (2.0f * (((float) M_PI) * logf(alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(-1.0) / Float32(Float32(2.0) * Float32(Float32(pi) * log(alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-1.0) / (single(2.0) * (single(pi) * log(alpha)));
end

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 95.2%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
6. Step-by-step derivation

7. Applied rewrites 65.6%

   \[\leadsto \frac{\color{blue}{-1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1} \]
8. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \frac{-1}{\left(\pi \cdot \color{blue}{\log \left(\alpha \cdot \alpha\right)}\right) \cdot 1} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{-1}{\left(\pi \cdot \log \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \cdot 1} \]

   3. pow2 N/A

      \[\leadsto \frac{-1}{\left(\pi \cdot \log \color{blue}{\left({\alpha}^{2}\right)}\right) \cdot 1} \]

   4. log-pow N/A

      \[\leadsto \frac{-1}{\left(\pi \cdot \color{blue}{\left(2 \cdot \log \alpha\right)}\right) \cdot 1} \]

   5. lower-unsound-*.f32 N/A

      \[\leadsto \frac{-1}{\left(\pi \cdot \color{blue}{\left(2 \cdot \log \alpha\right)}\right) \cdot 1} \]

   6. lower-unsound-log.f32 65.6%

      \[\leadsto \frac{-1}{\left(\pi \cdot \left(2 \cdot \color{blue}{\log \alpha}\right)\right) \cdot 1} \]

9. Applied rewrites 65.6%

   \[\leadsto \frac{-1}{\left(\pi \cdot \color{blue}{\left(2 \cdot \log \alpha\right)}\right) \cdot 1} \]

10. Taylor expanded in cosTheta around 0

    \[\leadsto \frac{-1}{\color{blue}{2 \cdot \left(\pi \cdot \log \alpha\right)}} \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \frac{-1}{2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)}} \]

    2. lower-*.f32 N/A

       \[\leadsto \frac{-1}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\log \alpha}\right)} \]

    3. lower-PI.f32 N/A

       \[\leadsto \frac{-1}{2 \cdot \left(\pi \cdot \log \color{blue}{\alpha}\right)} \]

    4. lower-log.f32 65.6%

       \[\leadsto \frac{-1}{2 \cdot \left(\pi \cdot \log \alpha\right)} \]

12. Applied rewrites 65.6%

    \[\leadsto \frac{-1}{\color{blue}{2 \cdot \left(\pi \cdot \log \alpha\right)}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))